Central Limit Theorem for empirical transportation cost in general dimension

TL;DR

This paper establishes a Central Limit Theorem for empirical transportation costs in any dimension, demonstrating Gaussian limits and providing insights into the stability and uniqueness of optimal transportation potentials.

Contribution

It introduces new results on the uniqueness and stability of optimal transportation potentials, leading to a CLT for empirical transportation costs in general dimensions.

Findings

CLT holds for empirical transportation cost under mild conditions

Limiting distributions are Gaussian and explicitly characterized

Results apply to general target probability measures in R^d

Abstract

We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on R d , with d $\ge$ 1. We provide new results on the uniqueness and stability of the associated optimal transportation potentials , namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.